Lithium ion batteries, nickel hydride batteries, and other rechargeable batteries have become increasingly important in recent years as vehicular power sources and as power sources for personal computers and mobile and portable electronic devices. In particular, the lithium rechargeable battery, which provides a high energy density at a low weight, is preferentially used as a high-output power source for installation in vehicles. In a lithium rechargeable battery, materials (the active material) capable of reversibly incorporating and discharging the lithium ion (Li ion) are provided at the positive and negative electrodes and charging and discharge are performed by the shuttling of the Li ion between the positive and negative electrodes. Lithium transition metal oxides that contain lithium and a transition metal element are a typical example of the active material (positive electrode active material) used in the positive electrode of such lithium rechargeable batteries. For example, the use is preferred of a lithium transition metal oxide that contains at least nickel (Ni) as the transition metal element (a nickel-containing lithium transition metal oxide) and that has a layered crystal structure. Patent Literature 1 is provided as technical literature related to the active material in lithium rechargeable batteries.